Condensate-induced transitions between topologically ordered phases

We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetry-breaking phase transitions which applies to phases with topological excitations described by quantum groups or modular tensor categories. This enables us to deal with phases whose quasiparticles have noninteger quantum dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional rational conformal field theories, and we find that there is a beautiful connection between quantum group symmetry breaking and certain well-known constructions in conformal field theory, notably the coset construction, the construction of orbifold models, and more general conformal extensions. Besides the general framework, many representative examples are worked out in detail.

Figure 1

(Color online) A geometry with an interface between two topological phases related by the proposed breaking mechanism. In the inner region there is an unbroken phase while the outer region is in a broken phase. The interface states correspond to the representations that are confined in the outer region [i.e., the 1 and 3 representations of $\text{SU}{\left(2\right)}_4$]. The states on the outer edge belong to representations that are not confined in the outer region [the 3 and the $3^{\ast }$ of $\text{SU}{\left(3\right)}_1$].

Figure 2

A schematic of quantum group symmetry breaking. After breaking we arrive at an intermediate algebra $\mathcal{T}$ that may have irreps which are in fact confined. The low energy effective theory of the condensed phase is based on the unconfined algebra $\mathcal{U}$.

Figure 3

The ribbon equation, relating fusion to topological spin, and monodromy.